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Permutations and Combinations

Competitive Mathematics · Algebra

7 Part2 worked examples16 practice questions

You have 3 shirts and 4 trousers. How many different outfits can you make? The shirt can be chosen in 3 ways and with each shirt a trouser in 4 — in all outfits. The whole chapter grows from this “multiply” idea. When order matters (a phone number, sitting in a queue) we count permutations; when we only pick a group and order makes no difference (a committee) we count combinations. So in every question first ask: “does changing the order give a new way?”

The two basic formulas
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The basic counting rules

The multiplication and addition rules

Multiply for step-by-step work; add for “or” alternatives; and the factorial.

Multiplication, addition and factorial

“and” = multiply, “or” = add

When stages follow one another the choices multiply; when it is one thing or another they add.

  • Multiplication rule: if one task can be done in ways and then another in ways, both can be done in ways.
  • Addition rule: if a task can be done either in the first manner ( ways) or the second ( ways) and not both, the total is .
  • Factorial: ; ; . objects can be lined up in ways.
  • If repetition is allowed (the same choices each time): ways. Without repetition: .
Example 1 · Basic
A hall has 9 gates. In how many ways can a man enter through one gate and leave through a different gate?
Solution, step by step
  1. 1There are ways to enter and ways to leave (a different gate).
  2. 2Multiplication rule:
Answer 72
Example 2 · Basic
In how many ways can a child with 4 pockets put 3 marbles into the pockets?
Solution, step by step
  1. 14 pockets for each marble (repetition is allowed).
  2. 2
Answer 64
BasicTry it yourself first
A hall has 12 gates. In how many ways can a man enter through one gate and leave through a different gate?
BasicTry it yourself first
There are 6 routes from city A to B and 7 from B to C. In how many ways can one go from A to C via B?
StandardTry it yourself first
A class has 18 boys and 12 girls. The monitor must be one boy or one girl. In how many ways?
BasicTry it yourself first
A child has 5 pockets and 3 marbles. In how many ways can the marbles be put into the pockets?
BasicTry it yourself first
4 books are to be given away to 6 boys, each boy being eligible for any book. In how many ways?
StandardTry it yourself first
A man has 5 friends to invite and 4 servants to carry the cards. Each card can be sent by any servant. In how many ways?
BasicTry it yourself first
An exam has 8 true/false questions. In how many ways can all of them be answered?
StandardTry it yourself first
6 matches are played in a cricket series (no ties or washouts). In how many ways can the results be declared?
StandardTry it yourself first
A railway line has 12 stations. How many different journey tickets are needed (A to B and B to A are different)?
StandardTry it yourself first
A 6-digit phone number has all different digits. If the digits at the extreme left and right are fixed as 4 and 9, how many such numbers are possible?
AdvancedTry it yourself first
An 8-digit phone number has all different digits. If the digits at the extreme left and right are fixed as 4 and 9, how many such numbers are possible?
StandardTry it yourself first
A flag with 4 vertical stripes is to be coloured using 5 colours so that no two adjacent stripes have the same colour. In how many ways?
StandardTry it yourself first
A flag with 5 vertical stripes is to be coloured using 3 colours so that no two adjacent stripes have the same colour. In how many ways?
AdvancedTry it yourself first
A flag with 4 vertical stripes is to be coloured using 6 colours so that no two adjacent stripes have the same colour. In how many ways?
AdvancedTry it yourself first
A flag with 5 vertical stripes is to be coloured using 4 colours so that no two adjacent stripes have the same colour. In how many ways?
AdvancedTry it yourself first
A flag with 3 vertical stripes is to be coloured using 7 colours so that no two adjacent stripes have the same colour. In how many ways?
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Permutations

When order matters

, letters of words, numbers from digits, seating in a row or at a round table.

and the formula

falling factors

The ways to pick of objects in order.

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Letters of words and numbers

Dividing for repeats and restrictions

Divide by the factorials of repeated letters; and conditions like vowels together/apart, at the ends, in odd places.

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Rows, round tables and necklaces

Fix one person

in a row; at a round table; for a necklace.

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3

Combinations

When you only choose

, committees, question papers, handshakes, lines and triangles.

and committees

Groups, not order

The value of , and committee / question-choosing problems.

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Handshakes, lines and triangles

Pairs and triples

One handshake per pair ; lines, triangles, quadrilaterals and diagonals from points.

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4

Mixed and clever problems

Paths, patterns and symmetry

Grid paths, “no two adjacent” patterns and symmetry arguments.

Grids, patterns and symmetry

Choose which steps carry the choice

Each move is one of two kinds: choose which of the total moves go north.

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5

Traps and fast tricks

Slips · tricks

Where marks go in permutation-combination questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Rules and checks

Habits for speed.

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6

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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7

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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